Math 204 EXAM 1

Observation

-Anecdotal notes -Checklists -Question probes

What should be assessed?

-Conceptual understanding and procedural fluency -Strategic Competence and Adaptive reasoning -Productive dispositions

Literature in Math

-Creates high-cognitive- demand tasks -Multiple tasks from one story -Resonates with children's experiences and imagination

Student Self-Assessment

-Exit slips -Open ended prompts -Tell students why they are doing the self-assessment -Promotes students to be active, rather than passive learners

High-Stakes Tests

-External tests need to be part of your assessment plan -Teach the big ideas in the math curriculum that are aligned to state and local standards -Identify the broad concpetual foundations in the standards and select problem-based tasks that foster this understanding

Problem-Based Task

-Focus on important mathematical concepts or skill -Stimulate connection of students' prior knowledge -Allow multiple solution methods and variety of tools -Opportunities for students to correct themselves along the way -Occasions for students to confront misconceptions -Encourage student reasoning and thinking -Opportunities to use mathematical processes and practice -Generate data for instructional making

What does it mean to DO Mathematics?

-Generating strategies for solving problems -Applying those approaches, seeing if they lead to solutions -Checking to see whether your answers make sense

Interviews

-Hearing students; descriptions and strategies -Probe their understanding -Identify strengths and gaps -Probing questions

Why didn't VA adopt Common Core?

-Helped develop common core -Already developed their own standards -Standards weren't high enough

Tips for Solving Problems:

-Searching for patterns -Analyzing situations -Generalizing relationships -Experimenting/Explaining -Rational understanding -Measuring with tools - manipulatives

Rubrics w/ a scale based on predetermined criteria

-Subjective -Shown to students ahead of time -Permits students to see what is central to excellent performance -Provides scoring guidelines for teacher to use in analyzing student performance

Problem Solving

-Teach for problem solving -Teach about problem solving -Teach through problem solving

Problem solving strategies

-Visualize -Look for patterns -Predict and check for reasonableness -Formulate conjectures and justify claims -Create a list, table, or chart -Simplify or change the problem -Write an equation

Grading

-a statistic to communicate to others an achievement level -accuracy and validity dependent on information used to generate the grade, professional judgment of the teacher and alignment of assessment with goals and objectives of the instruction -should be used with other information about a student's work (problem-solving process, attitudes, and beliefs)

Practice: new perspective Benefits/weaknesses

-develop conceptual ideas and more elaborate and useful connections -try out alternative and flexible strategies -encourage more students to understand -get the message that mathematics is about figuring things out

Why change the way we teach math?

-for our students to be competitive in the global market -so students can understand the complex issues they must confront as responsible citizens

Tests

-need to go beyond requiring students to demonstrate only procedural knowledge -tests need to be designed so they match the goals of the instruction

Worthwhile task

-no prescribed or memorized rules or methods to solve -do not have a perception that there is one "correct" solution -have the potential to provide new mathematical insights or knowledge

Importance of writing in math

-rehersal for classroom discussion -can serve as a written record that remains long after the lesson -focus students on the need to precise language in mathematics -written product provides evidence of student understanding -engages students in reflecting on what strategy makes sense

Productive Talk for Supporting Classroom Discussions (5)

-revoicing -rephrasing -reasoning -elaborating -waiting

Assessment

-should not be separate from instruction -Incorporate critical mathematical practices and processes -Summative assessments are cumulative -Formative assessments are to check student development

Assessment OF Learning Assessment FOR learning

-students are evaluated on what they know at a given moment -students are continually evaluated so instruction is targeted to gaps

Teach FOR problem solving

-teaching skills, then providing problems to practice those skills explain-practice-apply

Writing as an Assessment Tool

-writing integrates instruction and assessment -writing provides a window to student perceptions and thinking -Writing for early learners is drawing and markings -Writing is a script for student that have trouble verbalizing and thinking -Writing is a valuable tool to share with parents at conferences

The Before Phase (3)

1. Activate specific prior knowledge related to the learning goals 2. Be certain the problem is understood 3. Establish clear expectations

Flexible Grouping (6)

1. Allows collaboration on tasks 2. Provides support and challenges 3. Increases chance to communicate about mathematics 4. Size and makeup of small groups vary in purpose and strategic manner 5. Key to successful grouping is individual accountability 6. Establishing team building for shared responsibillity

NCTM Principles and Standards (2)

1. Assessment should enhance student learning 2. Assessment is a valuable tool for making instructional decisions

Tiered Lesson (4)

1. Degree of assistance- examples 2. How structured the task is- adaptations 3. Complexity of the task- tools to help 4. Complexity of the process- how much time

Three-Phase Structure for Lessons

1. Getting Ready 2. Students work 3. Class Discussion

Questioning Considerations (5)

1. High level of questioning 2. Procedural and conceptual understanding 3. Initiation-response- feedback 4. Who is thinking the answer (accountability for trying) 5. How you respond to an answer (use talk tool)

Eight Step Lesson Plan Process

1. Know your focused mathematical learning goals 2. Consider your students' needs 3. Select, design or adapt a worthwhile task 4. Design lesson assessments 5. Plan the before phase 6. Plan the during phase 7. Plan the after phase 8. Reflect and refine

The During Phase (4)

1. Let go! 2. Notice students' mathematical thinking 3. Provide appropriate support 4. Proved worthwhile extensions

What to tell students (3)

1. Mathematical conventions-symbols, terminology, labels 2. Alternative methods 3. Clarification or formalization of students' methods

Formative method types (3)

1. Observation 2. Interviews 3. Problem-Based Tasks

The After Phase (3)

1. Promote a mathematical community of learners working together and having productive discussions (respect) 2. Listen actively without evaluation and use this as a second chance to find out how students are thinking (wait time) 3. Summarize main ideas and make connections or lay groundwork for future problems (exit slips)

Assessment Standards for School Math (6)

1. The Mathematic Standard 2. The Learning Standard 3.The Equity Standard 4. The Openness Standard 5. The Inference Standard 6. The Coherence Standard

Scoring w/ a Four-Point Rubric

1: Unsatisfactory: Little Accomplished 2: Marginal: Partial Accomplished 3: Proficient: Substantial Accomplishment 4: Excellent: Full Accomplishment

Teach ABOUT problem solving (Polya)

4 step problem-solving process 1. understand 2. devise a plan 3. carry out the plan 4. look back

Curriculum Principle

A curriculum in is more than a collection of activities: -it must be coherent -focused on important math -articulated across grades -connections to daily-life

Differentiating Instruction

A lesson plan includes strategies to support the range of different academic background found in a classroom -content -process -product

Which of the following statements is true? A. Begin with mathematical goals, consider student needs, and use these factors to create the three- phase lesson plan B. Start with a worthwhile task, and then see which standards it addresses and plan the lesson around the task. C. Design assessments after a lesson is planned in order to assess what was planned for the lesson. D. Plan the phases of the lesson sequentially, starting with the before phase and ending with the after phase

A. Begin with mathematical goals, consider student needs, and use these factors to create the three- phase lesson plan -The beginning process of planning a lesson are the mathematical goals that must be considered in relationship to what students know and what their interests and strengths are. these two steps, in addition to finding a task and considering what you will have students do to show that they learned the goals, are important to know before thinking about the three phases of the lesson itself

Which of the following is not a formative assessment method? A. High-stakes assessment B. Interviews C. Tasks D. Observations

A. High-stakes assessment -The three main formative assessment categories are observation, interviews, and tasks. A high-stakes assessment is a summative assessment that usually takes the form of end-of-year tests.

Many factors influence how mathematics are taught in a school system and which mathematics are covered. What are some of the most influential factors? A. National and international testing results B. The age of the students and their geographic location C. The presence of a new classroom textbook and a new teacher D. The size and wealth of the school system

A. National and international testing results - The results of testing - at the local, national, and international levels strongly influence the direction of mathematics in education

All of the following statements about mathematical proficiency are true except which? A. Procedural fluency means being accurate and quick at using standard algorithms for each operation. B. A productive disposition, or a can-do attitude when it comes to solving a problem with no clear solution path, is one component of mathematical proficiency C. Conceptual understanding means having a flexible interconnected knowledge of a topic, such as ratios D. Mathematical proficiencies for students are described in the CCSS-M 8 Mathematical Practices

A. Procedural fluency means being accurate and quick at using standard algorithms for each operation. -Procedural fluency involves selecting the most efficient strategy in any given situation, but that is not always the standard algorithm, as the examples in this section illustrate. If students' experience are limited to learning only standard algorithms, they do not develop a collection of strategies and are not able to develop strategic competence, such as picking the best strategy for the particular problem

Which of the following statement about dots are true? A. Red dots (new ideas) are best added to a person's network of concepts by being connected to blue dots (existing ideas). B. Learning theory describes various characteristics of learning, providing clear direction on how to teach mathematics. C. Teachers already have complete webs of blue dots, which enables them to show students how those dots are connected. D. Some students will not have blue dots to connect to red dots, so those blue dots will have to be developed through instruction

A. Red dots (new ideas) are best added to a person's network of concepts by being connected to blue dots (existing ideas). - Having relational knowledge (well connected blue dots) is critical for understanding math, and includes conceptual understanding, procedural proficiency and other dispositions. Instruction that builds on existing ideas helps students to build this network. All students bring blue dots and all student' blue dots are different. As a teacher, you are still growing blue and red dots. the connections you can make within math, and between theory and practice, will inform and improve your ability as a teacher of mathematics.

Which of the following statements represents the equity principle? A. The message of high expectations for all is intertwined with every other principle B. Math today requires not only computation skills, but also the ability to think and reason C. Calculators and computers should be seen as essential tools for doing and learning mathematics D. Coherence speaks to the importance of building instruction around being ideas

A. The message of high expectations for all is intertwined with every other principle -The equity principle is embedded in all of the other standards, as expectations for all students must be high in order to address the other components

As you assess your students and learn about their strengths and weaknesses, the most important result is that you will be able to: A. target lessons that specifically address the students' naive understandings and misconceptions. B. identify the lowest-performing student in the class. C .give more summative tests. D. select a new mathematics textbook series.

A. target lessons that specifically address the students' naive understandings and misconceptions. -As you learn more about your students, you will be able to target lessons to address their naive understandings and misconceptions through the learning supports provided throughout this book. This is how the effective use of assessments shapes improved instruction for all students.

Strategic Competence

Ability to formulate, represent and solve mathematic problems

The Coherence Standard.

Assessment should be a coherent process.

The Openness Standard

Assessment should be an open process.

The Learning Standard

Assessment should enhance mathematics learning.

The Equity Standard

Assessment should promote high standards.

The Inference Standard.

Assessment should promote valid inferences about mathematics learning.

The Mathematic Standard

Assessment should reflect the mathematics that all students need to know.

Assessment Principle

Assessment should support the learning of important math and furnish useful information for teachers and students. -ongoing -student interaction & articulation

Which of the following is not a recommended teacher move in a three-phase lesson plan format? A. Sequencing student presentations of their strategies in an intentional manner. B. Illustrating how to solve a problem to ensure that students are ready to practice. C. Observing students and asking probing questions. D. Posing a simpler problem as a way to elicit prior knowledge.

B. Illustrating how to solve a problem to ensure that students are ready to practice. -Illustrating how to solve a problem denies students the opportunity to design and implement their own strategy. The three phase model is designed to encourage this higher-level thinking. The three other choices are among many teacher moves that support inquiry, reasoning, and sense making.

Although all are important, which of the following teacher characteristics is most essential to demonstrate in order to help students persevere, think to try other strategies, and check their answers to problems? A. Positive Attitude B.Persistence C. Reflective Disposition D.Lifelong learning

B. Persistence - Persistence is most closely aligned to the behaviors students will need to demonstrate in order to persevere and make sense of problems and their answers

Which of the following is the best way to pose a task to students so that they experience doing mathematics? A. Give an example of how to do a problem that is similar to the one you have selected, but slightly easier, so the students know what to do. B. Share the problem, ask students to explore, stop them to see how they are doing, and let them keep going. C. Teach the skill that is needed to solve the problem first, then give this problem as an extension. D. Have the students take the problem home and solve it, then bring it back to school to discuss

B. Share the problem, ask students to explore, stop them to see how they are doing, and let them keep going -To ensure the students get involved in doing mathematics, they would need to experience the problems much like you did. Choose problems for which students have the background, but to which all students (K-8) can apply meaningful and deep thinking with One Up, One Down. If the students are at home, you are not able to monitor, offer strategies, hear their thinking, or involve students in comparing strategies.

Which of the following reasons provides the best justification for why teaching through problem solving is effective for the struggling learner? A. Students do not enjoy or benefit from drill and would rather solve one challenging and interesting mathematical task. B. Students are able to pull from their knowledge base and use a strategy they like, which increases their chance of success and thereby motivates them to solve the problem. C. Students are able to hear other solution strategies, and so have access to strategies that they did not have to come up with on their own. D. There are fewer exercises when each exercise is more complex; the shorter list of problems is more attractive to struggling learners.

B. Students are able to pull from their knowledge base and use a strategy they like, which increases their chance of success and thereby motivates them to solve the problem -Each of the other answers might have a little bit of truth to them, but there is something not true in each. The key is to have tasks that fit the learner, as this increases students' chance of success, which in turn increases their willingness to attempt the task.

Learning center lessons may include all of the following except which? A. The lessons will entail careful observation in the during phase to see which stations or which concepts will be the focus of the after phase. B. The lessons may involve additional time in the after phase to share all solutions form each station. C. The lesson may require more time in the during phase with more tasks to do at this time. D. The lesson may require more time in the before phase to ensure that the station is understood.

B. The lessons may involve additional time in the after phase to share all solutions form each station. -Not all activities have to be discussed in the after phase. It is more important to ensure that the mathematical ideas that are the reason for the learning centers are developed. This may mean focusing the discussion on just one station or asking students to make connections across stations.

Three math tasks were shared in this section. Each of the tasks was an example of doing math because: A. it involved an open-ended problem. B. it required looking for patterns and order. C. it was fun. D. it engaged the learner in reasoning mathematically.

B. it required looking for patterns and order -All of these answer choices correctly describe the three tasks, but what makes these tasks examples of "doing mathematics" is the fact that students had to notice mathematical patterns and sense of those patterns. Looking for patterns and order was necessary to reach an answer.

Drill: new perspective Benefits/weaknesses

Benefits: -increased facility with a procedure already learned -a review of facts or procedures for retention Weaknesses: -focus on a singular method -exclusion of flexible alternative methods -false appearance of understanding -rule-oriented view of mathematics

Implications for Teaching Math

Build new knowledge from prior knowledge Provide opportunities to communicate about math Build opportunities for reflective thought Encourage multiple approaches Engage students in productive struggle Treat errors as opportunities for learning Scaffold new content Honor diversity

Which of the following is an example of a summative assessment? A. A diagnostic interview B. A problem-based task C. An end-of-unit test D. A checklist to evaluate student problem-solving ability

C. An end-of-unit test -Summative assessments are cumulative evaluations that might generate a single score, such as an end-of-unit standardized test that is used in school districts.

Which of the following statements would be counterproductive for a diagnostic interview? A. Diagnostic interviews are a means of getting in-depth information about an individual student's knowledge of and mental strategies about concepts. B.In a diagnostic interview, a student is given a problem and asked to explain how to solve it without assistance from the teacher. C. Diagnostic interviews provide an opportunity to teach students how to do mathematics in a one-on-one setting D. Diagnostic assessments can give teachers help in determining student misconceptions

C. Diagnostic interviews provide an opportunity to teach students how to do mathematics in a one-on-one setting - Diagnostic interviews are not teaching opportunities. Rather, they are used to collect data to inform next instructional steps.

Which of the following statements is the best definition for the term "school effects" as it pertains to mathematics education? A. Schools have no effect on students' leaning of math, and students' learning is completely determined by how they interact with math outside of the classroom B. Schools affect students' learning of mathematics by creating the daily schedule for when various content areas are taught C. For many students, school is the only place where they get to experience math. D. Some schools have better learning environments for mathematics than others

C. For many students, school is the only place where they get to experience math -Although students are often exposed to reading, current/historical events, and science, math is commonly experienced only in school This makes the teacher's role significant in students' learning and understanding.

Two kinds of rubrics include which of the following? A. Transparent and opaque B. Open-ended and closed C. Generic and task-specific D. Student-centered and teacher-centered

C. Generic and task-specific -Two kinds of rubrics include generic, which can be used for many different tasks or even subjects; and task-specific, which use indicators that are directly and tied to a particular task or assignment

One important function of rubric is what? A. It can be easy to create B. It acts like a test C. It permits students to see what excellent performance looks like D. It gives teachers an immediate grade

C. It permits students to see what excellent performance looks like -A rubric consists of a scale based on predetermined criteria. It has two important functions: (1) it permits the student to see what is central to excellent performance, and (2) it provides the teacher with scoring guidelines that support analysis of students' work

The accuracy of a student's grade depends most importantly on what? A. The number of standards addressed by a single test B. Whether one uses the mean or median of all the test scores C. Students' self-assessments move them to be active learners rather than passive learners. D. The length of the review period

C. Students' self-assessments move them to be active learners rather than passive learners. - The accuracy or validity of a grade is dependent on the information used in generating the grade, the professional judgement of the teacher, and the alignment of the multiple assessments with the stated goals and objectives.

Which of the following statements describing a test is true? A. Students' self-assessments should always be in the form of an open-ended writing prompt B. Students' self-assessments should be weighted more than other forms of teacher-evaluated assessments. C. Students' self-assessments move them to be active learners rather than passive learners. D. Students' self-assessments should focus on what they learned in previous years

C. Students' self-assessments move them to be active learners rather than passive learners. -By actively engaging in an evaluation of what they know and what they are still confused about, students become active partners in the learning process. This is an important lifelong skill.

Adaptive Reasoning

Capacity for logical thought, reflection explanation, and justification

Conceptual Understanding

Comprehension of mathematical concepts, operations, and relations

5 Strands of Mathematical Proficiency

Conceptual Uderstanding Procedural Fluency Strategic Competence Productive Disposition Adaptive Reasoning

Six Principles and Standards for School Mathematics

Curriculum Assesment Teaching Technology Learning Equity

In teaching through problem-solving, students are engaged in doing all of the following EXCEPT which? A.Determining their own solution path B. Making connections to other concepts and examples C. Convincing the teacher and their peers that their solution makes sense D. Checking their answers with others and the textbook to confirm that it is correct

D. Checking their answers with others and the textbook to confirm that it is correct -The correctness of a soution should lie in one's own justification, not in input from external sources such as the teacher, textbook, or other students. It is appropriate to have students share solutions for the purpose of critiquing each other's reasoning and making connections between strategies, but that is different from using other solutions to "check your answer"

Which of the following is not true? A. Practice is a meaningful engagement with mathematical ideas and should focus on helping students develop connections among mathematical ideas. B. Drill can provide an increased facility with a procedure that has already been learned. C. Worthwhile tasks can and should be used to teach both concepts and procedures. D. Developing procedural knowledge requires practice and drill.

D. Developing procedural knowledge requires practice and drill. -developing procedural knowledge can occur without drill. Students can master their facts and learn algorithms for the procedures they need to learn through meaningful practice and solving worthwhile task. Drill can support learning, but caution must be taken that the drill does not result in students blindly applying rules. Teachers should also not assume that students understand because they are able to imitate a procedure they have been shown.

Constructivism and sociocultural theories have implication for teaching. Which of the following teaching strategies would be "weak" in terms of helping students learn based on these theories? A. Showing students two different samples of student work in which the answers were different and discussing publicly which one is correct (or are both correct) B. Introducing multiplication by reading a children's book about arrays, such as 100 Hungry Ants C. Illustrating how to fill a ten frame and then asking students to share how many counters they see and how they see it D. Having students sort the facts that they know and then work on the facts that they do not know by trying to memorize them

D. Having students sort the facts that they know and then work on the facts that they do not know by trying to memorize them -Memorizing is not a method for students to connect their prior knowledge and build interconnected networks. However, sorting facts and identifying which ones still have their prior knowledge (for example, their known fact). If they know 5 + 5, how can they use that knowledge to solve an unknown fact, such as 5 + 7?

The purpose of differentiation is what? A. Provide choices for students. B. Provide more challenging tasks for more capable students. C. Provide a variety of tasks to enrich the after discussion. D. Proved tasks that are accessible and challenging for all students.

D. Proved tasks that are accessible and challenging for all students. -All of these answers have some truth to them, but the purpose of differentiating instruction is to ensure that every student is able to participate and be engaged in challenging math. Because students have different backgrounds, strengths, interests, experiences and knowledge, a task that is open, tiered, or parallel is more likely to provide that interest and challenge to all students

Which of the following statements is true about features of worthwhile tasks? A. Tasks that have multiple entry points mean that students have choices about which task they want to solve and they will have different answers based on the problem they chose. B. Worthwhile tasks are much like story problems because they are connected to real-life contexts. C. Students build up from tasks that are considered low level to ones that are considered high level over the course of a unit on a particular topic D. Relevant tasks includes ones that are interesting to students and that address important mathematical ideas; they may come from literature, the media, or a textbook

D. Relevant tasks includes ones that are interesting to students and that address important mathematical ideas; they may come from literature, the media, or a textbook -Relevant tasks can come from any source, but they must engage the learner in studying important mathematical ideas. At all ages and for all students, high-level tasks are appropriate from day one and on every day. The concept of multiple entry points is not about choice of tasks, but about different approaches to the same task. Worthwhile tasks can be in story form, but can also be purely symbolic. Conversely, story problems may be routine and thus not engage students in mathematical inquiry.

Which of the following is not on the list of important items to share with families related to mathematics? A. The role of technology in learning math. B. Why an inquiry is important in learning math. C. What a worthwhile task looks like. D. The amount of time their child should be drilling at home to ensure that they remember skills.

D. The amount of time their child should be drilling at home to ensure that they remember skills. -Parents need guidance on homework, but drilling is not the focus of the discussion. Instead, parents need to know how to support their children as the students engage in inquiry. This may include providing questioning prompts that help students with their reading of story problems or prompts to help them decide how to solve a problem.

What are the large groups in which Common Core divides the content expectations for students?

Domains

Teaching Principle

Effective math teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well. -provide strategies to enhance learning

Equity Principle

Excellence in math education requires equity -high expectations & strong support

Productive Disposition

Habitual inclination to see math as sensible, useful, and worthwhile coupled with a belief in diligence and one's own efficacy

Metacognitive skill development (ITHINK)

INDIVIDUALLY think about the task THINK about the problem HOW can it be solved? IDENTIFY a strategy to solve the problem NOTICE how your strategy helped you solve the problem. KEEP thinking about the problem. Does it make sense.

Parallel Tasks

Involves choice. Choice helps students become more self-directed learners

Constructivist Theory -Assimilation -Accommodation/Disequalibrium

Jean Piaget Learners are not blank slates but creators of their own knowledge Networks and cognitive schema are the products of constructing knowledge Reflective though is how people modify schemas to incorporate new ideas -new ideas fit within prior knowledge (blue dots) -new idea does not fit with existing knowledge

As a teacher...

Knowledge of Math Life-long Learning Readiness for Change Positive Attitude Persistence

Sociocultural Theory -Semiotic mediation

Lev Vygotsky Mental processes exist between and among people in social interactions Learning (working in their ZPD) - how beliefs, attitudes, and goals are affected by sociocultural practices and institutions

6 Content Standards VA

Number, Number Sense Computation and Estimation Measurement Geometry Probability & Stats Patterns, Function& Algebra

5 Content Standards (Common Core)

Numbers & Operations Algebra Geometry Measurement Data Analysis

The 5 Process Standards

Problem-solving Reasoning & Proof Representation Communication Connections

Bloom's Taxonomy

Promotion of higher-level thinking -Collaborate -Develop -Investigate -Formulate

Common Core State Standards

Puts a coherent vision of standards across the states @ specific grade levels (Math & English)

Continuum of Understanding - Rational Understanding -Instrumental Understanding

Richard Skemp -Knowing what to do and why -Doing something w/o understanding

Procedural Fluency

Skill in carrying out procedures flexibility, accurately, efficiently, and appropriately

Activity Evaluation and Selection Guide

Step 1- How is the activity done? Step 2- What is the purpose? Step 3- Can the activity accomplish your learning goals? Step 4- What must you do before?

Learning Principle

Student must learn mathematics with understanding -actively building new knowledge from experience & prior knowledge -individually & in groups

Technology Principle

Technology is essential in teaching and learning math -influences the math taught -enhances student learning

Multiple Entry Points Exit Points

The task can be approached in a variety of ways and has varying degrees of challenge within it. Various ways to express solution that reveal a range of mathematical sophistication and have the potential to generate new questions.

Teach THROUGH problem solving

teaching content through real context, problems, situations, models and exploration

Understanding

the measure of quality and quantity of connections between new ideas and existing ideas